Questions tagged [linear-pde]
Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.
371
questions
2
votes
0
answers
99
views
Question about the second order linear elliptic PDE on closed manifold
Recently I see a question
linear second order PDE
in which user Pedro post a reference in Gilbarg's book, which said that the solvability of the linear PDE
$$
\Delta u +B^{i}(x)u_{i}+C(x)u=f
$$
is ...
0
votes
1
answer
120
views
Riesz transform after linear transformation
I am encountering the term $\partial_x \mathcal{R}_x(f(x,y))$. I needed to do the following linear transformation
$$x' = a x+ by,\,\,\,\,\, y'=ax-by,\,\,\, and \,\,f(x,y)=g(x',y') $$
I ended up with ...
2
votes
0
answers
39
views
Estimates for higher order derivatives of the Airy Kernel
Consider the kdv equation (from here)
$$\left\{\begin{array}{l}
\partial_{t} v+\partial_{x}^{3} v=0 \\
v(x, 0)=v_{0}(x)
\end{array}\right.$$
Its solution can be written as $v(t,x)=S_t*v_0(x),$ where $...
3
votes
2
answers
243
views
Under which conditions does this PDE have unique solutions
Assume that $f:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ is smooth and consider the linear equation $$\mathrm{div}\, (u)(x) = f(x,u(x)),$$ where $u:\mathbb{R}^n \to \mathbb{R}^n $ is a smooth ...
4
votes
1
answer
218
views
Asymptotics of integral representation of distribution
I initially posted this question at MSE (here), but I have gotten no response, so I figured I would ask it to this community.
Background: I am studying the PDE $$\,\,\,\,\,\,\,\,\,\,\,\,i\partial_t \...
1
vote
1
answer
379
views
Explicit solution for a linear drift-diffusion equation (Fokker-Planck equation) on whole space
I'm wondering if there might be an explicit solution for the following linear PDE in two space dimensions $(x_1,x_2)$ on the whole space $\mathbb{R}^2$:
$$
\partial_t f = {div} \left [\left( \...
4
votes
0
answers
141
views
Uniqueness of the "weak solution" to Fokker-Plank PDE
Let $C_b^2(\mathbb R_+)$ be the set of functions $f: \mathbb R_+\to\mathbb R$ s.t. $f, f' ,f''$ are bounded and $f(0)=0$. Consider a measurable function $p: \mathbb R_+^2\to\mathbb R_+$ satisfying
$$\...
0
votes
1
answer
162
views
Explicit solutions for linear system of PDEs with constant coefficients
I've been recently trying to to solve the following system of linear 1st order PDE's:
$f:\Omega^d\xrightarrow{}\mathbb{R},\quad A^{(k)}\in\mathbb{R}^{N\times N},\quad B^{(k)}\in\mathbb{R}^N$
$\dfrac{\...
1
vote
0
answers
37
views
Parabolic theory for singular coefficients on bounded domains (Reference Request)
In Evans, the theory for linear PDE of parabolic type with bounded coefficients is developed. There are nice results such as long-existence of weak solutions and the parabolic regularity theorems.
Is ...
2
votes
1
answer
256
views
Using Darboux's to solve 2D system of first order linear PDEs with variable coefficients
I've spent some time over the last few days looking at the references suggested in this question and this question and I think the information therein is my best shot at solving this system that arose ...
1
vote
0
answers
138
views
Generalized functional for solution of PDEs
Asked this on Math Stack Exchange awhile ago but it got ignored then deleted.
To solve a differential equation of one variable, you need constraints equal to the number of derivatives.
For a partial ...
3
votes
1
answer
318
views
Maximum principle and linear transport
Let us consider the linear transport equation
$$
\partial_t u + \mathrm{div}(a(t,x)u)=0
$$
with initial data $u(0,\cdot) = u_0$ in $\mathbb R^N$.
Here we consider a smooth Lipschitz vector field $a$.
...
2
votes
1
answer
121
views
Spectral analysis for nonlocal elliptic operator
Suppose $\Omega\subset\mathbb{R}^3$ is a bounded domain with smooth boundary. We note by $(-\Delta)^{-1}$ the inverse Laplacian i.e. $f\mapsto u$ where $u$ is the unique solution to
$$-\Delta u=f,\...
1
vote
0
answers
116
views
Liouville theorem for an elliptic equation with gradient perturbation
How can I prove the following Liouville theorem for an elliptic equation with gradient perturbation?
Let $u \in L^2(\mathbb R^n;\mathbb R)$ be a smooth solution of
$$ -\Delta u + v \cdot \nabla u = 0 ...
1
vote
0
answers
47
views
Can we find a uniform bound of the solution of a series of linear partial differential equations related to a parameter
Let $\sigma \in[0,1]$,we consider following series of linear partial differential equations related to the parameter $\sigma$,for example
$$
\left\{\begin{aligned}
\Delta \Phi &=\sigma f(x, y) \...
2
votes
0
answers
58
views
Decay of solution for linear system with damping
Let us consider the following linear system with damping:
$$
\begin{cases}
u_t - u_x = -\frac{1}{2} (u+v)\\
v_t + v_x = -\frac{1}{2} (u+v)
\end{cases}
$$
Let's write the solution as $w=(u,v)$ ...
2
votes
0
answers
112
views
Solve a coupled PDE in a rectangle
We consider a coupled PDE in a rectangle $\Omega=(-1,1)\times(-1,1)$. For the simplicity, we assume that the functions are periodic in $x_{1}$ direction.
\begin{cases}
\nabla\cdot u=f_{1},\ & \...
0
votes
0
answers
65
views
Regularity of solution to Cauchy problem given regular initial data
Let $f\in L^2_1([0,T]\times \mathbb{T}^m)$ (Sobolev space of maps of regularity $1$, $\mathbb{T}^m$ is the $m$-dimensional torus)
be a solution of a Cauchy problem
$$\frac{d}{dt} f(t) = A f(t)$$
$$f(0)...
1
vote
0
answers
115
views
Reference for global theory of Schrödinger operators
Question. What is a good reference to learn about the spectral properties of Schrödinger operators in $\mathbf{R}^n$? I am specifically interested in references that discuss examples where the ...
11
votes
2
answers
632
views
A singular differential equation
In a neighbourhood of $0$ in $\mathbb{R}^n$ a smooth function $h=h(x)$, $h(0)=0$, is given. Take arbitrary real numbers $w,\lambda_1,\dots,\lambda_n\in\mathbb{R}$.
The problem is to find a smooth ...
0
votes
0
answers
169
views
Has this form of the heat equation been solved for the radiation boundary condition
Below is a solution to a special form of the heat equation. I have found the postings on the heat equation and they are far above my head. I tried to find a tag on Transport Theory for both heat and ...
2
votes
2
answers
124
views
Density of traces of solutions to an elliptic equation
Let $D_1$ be a domain with smooth boundary and assume that $D_1$ is a proper subset of $D_2$ which is itself a bounded domain in $\mathbb R^n$ with a smooth boundary. Assume also that $D_2\setminus ...
3
votes
0
answers
93
views
How to find a particular solution of a non-homogeneous parabolic partial differential equation
Consider the following non-homogeneous parabolic partial differential equation (PDE)
\begin{align}
\left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma}{r} \sin\psi \frac{\partial}{\partial \...
4
votes
2
answers
248
views
spaces of smooth functions for linear hyperbolic PDE
Which classes of (scalar or systems of) linear first or second order hyperbolic PDEs $Lf=g$ in $n$ variables and spaces $S$ of functions have the property that there is a retarded Green's function $G:...
2
votes
1
answer
143
views
Existence of a global analytic solution to a linear first order PDE
Let $B=\lbrace \|z\|<1\rbrace$ be a unit ball in $\mathbb{C}^n, n\geq 2.$ Let
$f_1,\cdots, f_n, f$ be holomorphic functions on $B.$ Now, consider the following
first order, linear PDE:
$$f_1\...
6
votes
1
answer
245
views
Space of solutions to a fourth order wave equation
I'm interested in finding solutions a fourth order version of the standard wave equation in $d$ dimensional Minkowski spacetime $\mathcal{M}^d$. Defining $\Box := \partial_0^2 - \sum_{i = 1}^{d-1} \...
5
votes
2
answers
260
views
Linear hyperbolic PDE on compact two dimensional domain
Consider the equation
$$
\begin{equation}
\frac{\partial^2f}{\partial x\partial y}=f
\end{equation}
$$
on a Jordan domain (i.e. the interior of a simple, closed curve on the plane). The equation is ...
3
votes
1
answer
185
views
Linear transport equation with Lipschitz conditions
Given the equation here, I would like to ask the following relaxed question:
Consider the PDE
$$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = ...
5
votes
2
answers
347
views
Linear transport equation with unbounded coefficients
Consider the PDE
$$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = f_0(x) \in \mathscr S(\mathbb R^n).$
I am wondering then if $q$ and all its ...
3
votes
1
answer
262
views
Is this equation of hyperbolic type?
I want to now whether this equation is of hyperbolic type:
$$(1-\partial_{xx})y_{tt}+y_{xxxx}=0$$
with boundary conditions $$y(t,0)=y(t,1)=y_x(t,0)=y_x(t,1)=0.$$
I would say that the answer is yes. By ...
5
votes
0
answers
196
views
Where to locate $0\in \Omega$ to get $u_{\varepsilon}(0)\neq 0$ where $\Delta u_{\varepsilon} + (\lambda-\varepsilon) u_{\varepsilon} = \frac{1}{|x|}$
Let $\Omega \subset \mathbb{R}^3$ a smooth bounded domain with $0\in \Omega$ and $u_\varepsilon(x)$ the solution to
$$
\Delta u_\varepsilon + (\lambda-\varepsilon) u_\varepsilon = \frac{1}{|x|}\quad \...
2
votes
1
answer
263
views
Even and odd solutions for the Schrödinger equation
We consider $2a$ - periodic smooth solutions for
\begin{eqnarray*}
-\Delta u+V(x)\,u=0\qquad\hbox{in}\:[-a,a]
\end{eqnarray*}
We assume that $V$ is smooth and even (i.e. $V(-x)=V(x)$). We also assume ...
2
votes
0
answers
94
views
Evolution PDE in dual space : Generalization of a result of Gelfand
The following result is proved in "Generalized functions", Volume 3, Chapter 2.2 by Gelfand :
Let $\Phi$ be a Fréchet space with dual space $\Phi'$ endowed with the weak topology. For ...
1
vote
0
answers
104
views
How can you make a PDE solution stabilize over time?
this is a problem I have been thinking about lately. I tried asking on stack exchange as well but did not find an answer.
Suppose I have a simple linear first order PDE of the form:
$$au_x+bu_y=0$$
I ...
6
votes
0
answers
139
views
Does $f \in L^1([0,T]; S'(\mathbb R^n))$ define a $(1+n)$-dimensional distribution?
Let $f : [0,T] \rightarrow S'(\mathbb R^n)$ be a family of tempered distributions satisfying
$$\langle f(t), \phi \rangle \in L^1([0,T])$$
for any Schwartz function $\phi \in S(\mathbb R^n)$.
Does $f$ ...
2
votes
1
answer
252
views
How to trap a particle without using potential field which is infinity at some point? (quantum physics) If impossible, how to prove it?
As we all know, the wave function of the stationary state a quantum particle trapped in a rigid box (with infinite potential outside the box) cannot have a non-zero value outside the box. So can we ...
1
vote
1
answer
103
views
Relation between the solutions $v_t=Lv$ and $v_t=Av$ if $A$ is a relatively compact perturbation of the linear operator $L$
In a nutshell, here is my question. I read and know about the relation between the spectra of $L$ and $A$ if $A$ is a relatively compact perturbation of $L$. However, for my purpose, I am interested ...
2
votes
1
answer
151
views
What rotations are used as a reduction step in Kenig-Ruiz-Sogge's uniform Sobolev estimate?
I think I have understood the bulk of the paper [KRS], but one of the parts I cannot understand is when the authors reduce Theorem 2.1 (p.332) into Proposition 2.1 (p.335). I can understand all the ...
0
votes
1
answer
96
views
Reference request: Is if possible to estimate the local behaviour of the solution of $\nabla \cdot a(x) \nabla f=g$ via constant coefficients?
Consider the divergence form uniformly elliptic operator $\nabla \cdot a(x) \nabla$
where the coefficient $a$ are smooth and bounded and $D$ is a bounded
and smooth domain of $\mathbb R^d$
$$
\begin{...
5
votes
2
answers
383
views
General solution to an ultrahyperbolic PDE
$\DeclareMathOperator\SO{SO}$The following PDE defined on $\mathbb{R}^2$ $$\frac{\partial}{\partial x}\frac{\partial}{\partial y}f(x,y) = 0,$$ has solution $$f(x,y) = g(x) + h(y),$$ where $g,h : \...
6
votes
2
answers
334
views
Vacuum region with positive measure for the Schrödinger equation
Let us consider the free Schrödinger equation $(i\partial_t+\Delta_x)\psi=0$ in $\mathbb{R}_t\times\mathbb{R}_x^d$. I'm trying to understand the structure of the vacuum region
$$\Omega(\psi):=\{(t,x)\...
2
votes
1
answer
237
views
Existence of divergence-free unit vector field in conformally rescaled euclidean metric
Question. Let $\Omega \subset \mathbf{R}^2$ be a convex polygonal domain, equipped with a Riemannian metric $g$. Under which conditions on $g$ is there a vector field $X$ in $\Omega$ with $\mathrm{div}...
2
votes
0
answers
113
views
Divergence-free constraint for a boundary integral equation
Consider the system
$$
\begin{cases}
\operatorname{curl} \operatorname{curl} \mathbf{u} = 0 \qquad B \cup (\mathbb{R}^3 \setminus \overline{B}) \\
c_1 (\operatorname{curl} \mathbf{u} \times \mathbf{n})...
3
votes
1
answer
258
views
References for Green functions of $\nabla \cdot a \nabla$ on a domain with $a \in L^\infty$
I am looking for a reference for basic properties of the Green function for a symmetric, uniformly elliptic operator $\nabla \cdot a \nabla$ where the coefficients $a_{ij}= a_{ji}$ are only assumed to ...
2
votes
0
answers
143
views
Explicit computation of a norm in context of operator-semigroups and differential equations
I am interested in the explicit calculation of the following norm $\vert \cdot \vert$.
Let $X$ a Banach space with norm $\Vert \cdot \Vert$ and $(T(t))_{t \geq 0}$ a strongly continuous one-parameter ...
2
votes
1
answer
162
views
The only rotation fields satisfying this PDE are constant
$\newcommand{\div}{\operatorname{div}}$$\newcommand{\SO}{\operatorname{SO(2)}}$$\newcommand{\R}{\operatorname{\mathbb{R}}}$$\newcommand{\bdx}{\partial_x}$$\newcommand{\bdy}{\partial_y}$$\newcommand{\...
2
votes
2
answers
236
views
Heat flow derivative of entropy
In a 1966 paper (Speed of Approach to Equilibrium for Kac's Caricature of a Maxwellian Gas, Arch. Rational Mech. Anal., Vol. 21), McKean seems to suggest that the successive derivatives of entropy $H (...
3
votes
1
answer
326
views
Neumann/Robin Laplacian semigroup well-known estimate
Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacian defined on:
$$D(\Delta_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \...
1
vote
0
answers
215
views
Fourier Transform; half space baby problem (new)
This question is related to a prior question i asked, see Fourier Transform ; half space elliptic baby problem.
Essentially I am asking the same question now but taking a lot more care.
So lets ...
4
votes
2
answers
217
views
Looking for a reference or the procedure on how to solve the parabolic equation with $L^2$-weight
Let $\zeta, u_0\in L^2(\Omega)$, with $\zeta \geq 0$ and $\Omega\subset \Bbb R^d$ open and bounded.
\begin{equation}\label{Star-3.7}
\begin{cases}
\partial_t u -\Delta u + \zeta u=0 &\mbox{ in }\...